Electric field controlled second-order anomalous Hall effect in altermagnets

TL;DR

The paper addresses how to realize and control a second-order anomalous Hall effect in altermagnets with zero net magnetization by leveraging electric-field driven quantum geometry. It introduces a minimal 2D Rashba-coupled hybrid altermagnet model that mixes $d_{x^2-y^2}$ ($B_{1g}$) and $d_{xy}$ ($B_{2g}$) orders via a tunable parameter $\alpha$ and includes Rashba spin-orbit coupling, enabling a field-induced Berry curvature through the Berry connection polarizability. The authors show that an external dc field generates a field-induced Berry curvature and a Berry curvature dipole $\mathbf{D}^{\mathrm{E}}(\theta)$, producing a second-harmonic Hall current $j^{2\omega}$ whose magnitude and angular dependence $\chi^{\text{AH}}(\theta,\phi)$ are controllable by $\alpha$, $\mu$, and RSOC $\lambda$, and can distinguish between pure altermagnetic orders. This all-electrical control and the sensitivity to the underlying altermagnetic symmetry offer a route to tunable nonlinear transport and spintronic applications, with candidate realizations in RuO$_2$ and MnTe at interfaces that break inversion symmetry.

Abstract

Altermagnets are a recently discovered class of compensated magnets with momentum-dependent spin splittings and unusual transport properties, even without a net magnetization. In the presence of combined four-fold rotation and time-reversal ($C_4\mathcal{T}$) symmetry, linear and also second-order, driven by a Berry curvature dipole, anomalous Hall responses are forbidden in any pure $d$-wave altermagnet. Nevertheless, here we find that the nontrivial quantum metric of the occupied Bloch states allows for an electric field induced Berry curvature dipole, which generates a strong and tunable second-order Hall current, enabling it to be switched on or off by simply adjusting the relative orientation between the symmetry-reducing dc field and the ac probe field. Specifically, we investigate the electric field induced second-order anomalous Hall response in a two-dimensional Rashba-coupled hybrid altermagnet that interpolates between $d_{x^2-y^2}$ ($B_{1g}$) and $d_{xy}$ ($B_{2g}$) altermagnet symmetry, motivated by recent proposals for mixed-symmetry states. Crucially, the nonlinear signal is highly sensitive to the underlying symmetry of the altermagnetic order at specific doping levels, offering a purely electrical method to distinguish distinct altermagnetic orders. Our results position hybrid altermagnets as a promising platform for controllable nonlinear transport and spintronic applications.

Figures (13)

• Figure 1: Berry connection polarizability (BCP) tensor components for one of the band: (a) $G^{1}_{xx}$, (b) $G^{1}_{xy}$, and (c) $G^{1}_{yy}$ in the first Brillouin zone. Field-induced Berry curvature for the same band , $\Omega^{\textrm{E}}(\mathbf{k})$, for dc-field orientations (d) $\theta=0$, (e) $\pi/4$, and (f) $\pi/2$, with range arrows indicating electric field directions. Calculations performed with $t=1.0$, $t_{\text{am}}=0.5 t$, $\alpha=0.5$, $\mu=0.3t$, and $\lambda=0.08t$.
• Figure 2: Momentum distribution of the Fermi-function-weighted derivatives of the field-induced Berry curvature in log-scale for the two bands, (a) $f_0~\partial_{k_x}\Omega^{\mathrm{E}}_1$, (b) $f_0~\partial_{k_y}\Omega^{\mathrm{E}}_1$, and for band 2, (c) $f_0~\partial_{k_x}\Omega^{\mathrm{E}}_2$, (d) $f_0~\partial_{k_y}\Omega^{\mathrm{E}}_2$, evaluated at $\theta = \pi/4$. Parameters are the same as in Fig. \ref{['fig:fig1']}.
• Figure 3: Second-order Hall conductivity $\chi^{\mathrm{AH}}$ as a function of angle $\phi$, set by the ac driving field $\mathbf{E}^\omega$, for several different angles $\theta$, set by the symmetry-breaking static field $\mathbf{\mathbf{E}}^{\mathrm{dc}}$, with the $x$-axis. $\chi^{\mathrm{AH}}$ is scaled by the prefactor $\mathcal{A}=\frac{e^3 \tau}{2 (1 + i \omega \tau) \hbar^2}$ and the parameters are the same as in Fig. \ref{['fig:fig1']}.
• Figure 4: Second-order Hall conductivity $\chi^{\mathrm{AH}}$ as a function of $\alpha$ for different RSOC $\lambda$ values and field orientations: (a) $\theta=0$, $\phi=\pi/2$ and (b) $\theta=0$, $\phi=\pi$ with $t = 1$, $t_{\text{am}}=0.5 t$, and $\mu=0.3t$ (same as Fig. \ref{['fig:fig1']}).
• Figure S1: Berry connection polarizability (BCP) tensor components for one of the band : (a) $G^{1}_{xx}$, (b) $G^{1}_{xy}$, and (c) $G^{1}_{yy}$ in the first Brillouin zone. Field-induced Berry curvature for the same band , $\Omega^{\textrm{E}}(\mathbf{k})$, for dc-field orientations (d) $\theta=0$, (e) $\pi/4$, and (f) $\pi/2$, with orange arrows indicating electric field directions. Calculations performed with $t=1.0$, $t_{\text{am}}=0.5 t$, $\alpha=1.0$, $\mu=0.3t$, and $\lambda=0.08t$.